In another thread, Lou Talman and Robert Hansen are discussing sequence.

Lou brings up the point that the ancient Greeks invented much of whatwe call mathematics without being especially proficient in arithmeticin the way our own subcultures were recently trained. We aren't thesame way about arithmetic today, since calculators and other devicesmade everything so easy, although we still school in "the algorithms"in ways the Greeks never did. Were they "unsuccessful in math"?

Quoting Lou:"""And the ancient Greeks---who invented modern mathematics---arecertainly a counterexample to your "natural progression". Theyaccomplished a great deal without beginning with the algorithms we askkids to study today. Indeed, it's likely that they weren't very goodat arithmetic at all. So their "progression", if there was such athing, was entirely different from the one you think you'veidentified.

This last example suggests very strongly that arithmetic, while it maybe *an* entry into mathematics, is not the *only* entry. Your "natural progression" completely ignores a significant possibility: Theprimacy of arithmetic is simply an artifact of a curriculum thatdenies entry to those who haven't acquired proficiency at arithmetic.(A curriculum, moreover, that's now strongly distorted by the effectsof fifty years of standardized, multiple-guess, truth-or-consequences,mis-matching tests.)"""

Of course once you get through whatever sequence growing up, you'renot done with sequences. It's always "one damn thing after another"(Henry Ford, on history). So even if we argue about the "one rightsequence" (I'm against the notion) for child-to-adult math learning,we're not done. What else might we try with adults?

I'm probably more into adult guinea pigging than most. For me, it'snot all about what we teach to 10 year olds.

Even with kids though, I emphasize V + F == E + 2. I don't worryabout what if it has more holes too much. It's a wire frame to beginwith probably and a simple convexity, a polyhedron for which we have aname probably, or part of a series.

[ Waterman Polyhedra for example, I worked on them, with Steve, theirdefiner and early sculptor of their form (he used spreadsheets andactual physical models made of little balls) -- our team onSynergetics-L (e.g. Gerald de Jong, myself) did a bunch of thecomputer graphics, using Qhull. Other yet more expert implementationsby other collaborators (with Steve Waterman) came later. Google andye shall find. ]

Lou again:"""And consider the popularity of puzzles like sudoku---which are basedon very mathematical, but non-arithmetic, reasoning---in a nation thatdespises mathematics. Where do such phenomena fit in your "naturalprogression"?"""